The eight corners of a cube are numbered 1 to 8 (as an example 'Diagram 1'). Its six face sums(the totals of the numbers on the four vertices of each face) are equal to 10, 14, 18, 22 and 26.

Diagram 2 displays the cube with the same information in a adifferent way. Also, five of the face sums are shown, in bold. The 'Front' face sum of 14 is not shown, but can be found by adding the numbers on the four outer vertices in this diagram.

An example of an opposite vertex pair in this numbered cube is 1/7.
A numbered cube is prime-faced if all six face sums are prime numbers.
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QUESTIONS:
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a) Draw a diagram showing a prime faced cube with more than two different face sums
Find a prime faced cube with opposite vertex pairs 1/2, 3/4, 5/6, 7/8.
c) Some prime numbers arise as a face sum of a prime faced cube and others never do. Find the ones that are possible; your answer must, in particular, make it clear why numbers not in your list are impossible.
d) Show that the opposite vertex pairs of a prime cube are one odd and one even.

June 10th 2006, 07:12 AM

CaptainBlack

Please post new questions in new threads, it makes it easier
for everyone to follow what is going on.

Also in questions which refer to diagrams it is advantageous if we
can see the diagrams, even though with a bit of effort we can
often reconstruct what they must have been :(